Epimorphism into Projective Object Splits

Theorem

Let $\mathbf C$ be a metacategory.

Let $P \in \mathbf C_0$ be a projective object of $\mathbf C$.

Let $e: E \twoheadrightarrow P$ be an epimorphism.

Then $e$ is a split epimorphism, i.e. it admits a retraction $f: P \to E$.

Proof

Consider the identity morphism $\operatorname{id}_P: P \to P$.

By definition of projective object, we obtain the following commutative diagram:

$\quad\quad \begin{xy} <0em,0em>*+{P} = "P", <4em,0em>*+{P} = "P2", <4em,4em>*+{E} = "E", "P";"E" **@{-} ?>*@{>} ?*!/_.6em/{f}, "P";"P2" **@{-} ?>*@{>} ?*!/^.6em/{\operatorname{id}_P}, "E";"P2" **@{-} ?>*@2{>} ?<>(.7)*{\vee} ?*!/_.6em/{e}, \end{xy}$

where $f = \bar{\operatorname{id}_P}$.

It follows that $e \circ f = \operatorname{id}_P$, and so $e$ is a split epimorphism.

$\blacksquare$

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 2.1.1$